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A374932 Number T(n,k) of partitions of [n] such that the maximal block element sum equals k; triangle T(n,k), n>=0, n <= k <= A000217(n), read by rows.

%I A374932 #31 Aug 02 2024 17:41:00
%S A374932 1,1,1,1,2,1,1,1,3,3,3,3,1,1,1,6,6,8,9,9,5,3,3,1,1,1,12,14,20,31,26,
%T A374932 32,19,14,11,10,5,3,3,1,1,1,26,31,59,78,111,108,113,76,67,57,39,39,21,
%U A374932 16,12,10,5,3,3,1,1,1,57,84,140,260,321,458,427,500,326,300,284,229,182,159,107,79,64,46,41,23,17,12,10,5,3,3,1,1,1
%N A374932 Number T(n,k) of partitions of [n] such that the maximal block element sum equals k; triangle T(n,k), n>=0, n <= k <= A000217(n), read by rows.
%H A374932 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A374932 T(5,7) = 8: 124|3|5, 12|34|5, 13|25|4, 14|25|3, 15|2|34, 1|25|34, 1|2|34|5, 1|25|3|4.
%e A374932 T(6,6) = 12: 123|4|5|6, 12|3|4|5|6, 13|24|5|6, 13|2|4|5|6, 14|23|5|6, 15|23|4|6, 1|23|4|5|6, 14|2|3|5|6, 15|24|3|6, 1|24|3|5|6, 15|2|3|4|6, 1|2|3|4|5|6.
%e A374932 T(6,7) = 14: 124|3|5|6, 12|34|5|6, 13|25|4|6, 16|23|4|5, 14|25|3|6, 16|24|3|5, 15|2|34|6, 16|25|34, 1|25|34|6, 16|2|34|5, 1|2|34|5|6, 16|25|3|4, 1|25|3|4|6, 16|2|3|4|5.
%e A374932 Triangle T(n,k) begins:
%e A374932   1;
%e A374932      1;
%e A374932         1, 1;
%e A374932            2, 1, 1,  1;
%e A374932               3, 3,  3,  3,  1,  1,  1;
%e A374932                  6,  6,  8,  9,  9,  5,  3,  3,  1,  1,  1;
%e A374932                     12, 14, 20, 31, 26, 32, 19, 14, 11, 10, 5, 3, 3, 1, 1, 1;
%e A374932                     ...
%Y A374932 Row sums give A000110.
%Y A374932 Main diagonal gives A375099.
%Y A374932 Number of terms in row n is A000124(n-1) for n>=1.
%Y A374932 Reversed rows converge to A294617.
%Y A374932 Cf. A000217.
%K A374932 nonn,tabf
%O A374932 0,5
%A A374932 _Alois P. Heinz_, Aug 01 2024